Even though this now links to the FAQ, there's one thing I want to highlight: awarding bounties and accepting answers have been completely independent of each other for a long time.
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Pops♦Apr 5 '12 at 14:24

1 Answer
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You can give the bounty to any (other) answer to your question. If your answer is the first fully correct solution to your issue, but other answers made progress towards it or contributed to you solving the problem, award the bounty to one of those answers. Otherwise, just let it expire.

Once you offer a bounty, the reputation is taken from you and can not come back.

You can not award yourself a bounty (otherwise, you could offer a bounty to get attention and good answers and upvotes, then return the rep to yourself, which would be abuse).

Awarding a bounty is not required. If nobody else made worthwhile contributions, let it expire. If anyone seems to have put in significant effort and made meaningful contributions, then they are by all means entitled to the reward you offered to draw them in.

One thing I've seen done is to have another user put a dummy post to accept the bounty if the question doesn't get answered--and the user will later give the bounty to whoever answers.
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ManishearthApr 5 '12 at 5:55